# Doesn't the Pareto-extension rule invalidate Eliaz's (2004) unified theorem of social choice?

Eliaz (2004) uses social aggregators to provide a unique "meta-theorem" from which Arrow and Gibbard-Satterthwaite follow as corollaries. He defines social aggregators as follows. Let $$\mathcal{P}$$ denote the set of all n-tuples of linear (thus strict) orderings over the elements of set $$X$$, where $$|X| \geq 3$$, and $$\mathbf{R}$$ a set of binary relations on $$X$$. A social aggregator is a function $$F: \mathcal{P} \rightarrow \mathbf{R}$$. A social welfare function is a social aggregator satisfying acyclicality, completeness, and existence of a best alternative; a social choice function takes more work to define, but in short, completeness is only required for at least one alternative which is socially ranked at least as high as all others.

Eliaz then proves that no non-dictatorial social aggregator can satisfy acyclicality, existence of a best alternative, weak Pareto efficiency, and "preference reversal", which is basically an independence requirement by which a reversal of the social relation must follow the same reversal in the preferences of an individual. He then shows that from the conditions above follow all the conditions in Arrow and Gibbard-Satterthwaite, which are thus just special cases of the general meta-theorem.

I have two key questions:

1. To be a social welfare function à la Arrow, do we not require more than just acyclicality - in particular, transitivity? Is it correct that what Eliaz calls a social welfare function, is actually a social decision function?

2. If so, isn't Sen's (1969) Pareto-extension rule a counter-example to the meta-theorem? For strict individual orderings, the Pareto-extension rule works as follows: all conflicting profiles (i.e. all those profiles in which there are two $$x,y \in X$$ and two $$i,j \in N$$, $$N$$ being the set of individuals, such that $$xP_i y$$ and $$yP_j x$$), then $$xRy$$ and $$yRx$$, $$R$$ being the social relation. In other words, the Pareto-extension rule resolves all conflicts by ties. This rule should satisfy all condition imposed by Eliaz: acyclicality (it is actually quasi-transitive, a stronger condition), existence of a best alternative, non-dictatorship, independence (subsumed by preference reversal), and clearly, weak Pareto efficiency.

Acyclicality as defined by Eliaz is simply the condition that the values of a social aggregator are transitive. It is different from the usual acyclicity condition according to which a relation is acyclic iff it has irreflexive transitive closure. The formulation of Eliaz is a bit cryptic, but he actually shows in his Observation 1 that his acyclicality implies transitivity. The converse is also easy; the simple details are below. The result of the Pareto extension rule need of course not be transitive.

Formally, acyclicality as defined by Eliaz says that for all $$R\in\mathbf{R}$$ and for every three alternatives $$a$$, $$b$$ and $$c$$ in $$A$$, if $$aRb$$ and $$\neg(cRb)$$, then $$\neg(cRa)$$.

But the condition that for every three alternatives $$a$$, $$b$$ and $$c$$ in $$A$$, if $$aRb$$ and $$\neg(cRb)$$, then $$\neg(cRa)$$ is simply the transitivity of $$R$$, formulated in a slightly weird but equivalent way.

Here is the simple proof: Assume that $$R$$ is transitive and $$aRb$$ and $$\neg(cRb)$$. If it were the case that $$cRa$$, then $$cRb$$ by transitivity, contradicting the assumption. For the other direction, assume that the condition holds and that $$cRa$$ and $$cRb$$ holds. We want to show that $$cRb$$. Indeed, if not then by the condition $$\neg(cRa)$$, contradicting the assumption.